Sign in adjoint derivative calculation

[spenrich]

I've been following the paper "Differentiating Through a Cone Program" and the code side-by-side, and I'm having trouble figuring out if there is a sign error in the adjoint derivative code or if I've misunderstood something.

diffcp/diffcp/cone_program.py

Lines 341 to 357 in 83080bc

	dw = -(x @ dx + y @ dy + s @ ds)
	dz = np.concatenate(
	[dx, D_proj_dual_cone.rmatvec(dy + ds) - ds, np.array([dw])])
	if np.allclose(dz, 0):
	r = np.zeros(dz.shape)
	elif mode == "dense":
	r = _diffcp._solve_adjoint_derivative_dense(M, MT, dz)
	else:
	r = _diffcp.lsqr(MT, dz).solution
	values = pi_z[cols] * r[rows + n] - pi_z[n + rows] * r[cols]
	dA = sparse.csc_matrix((values, (rows, cols)), shape=A.shape)
	db = pi_z[n:n + m] * r[-1] - pi_z[-1] * r[n:n + m]
	dc = pi_z[:n] * r[-1] - pi_z[-1] * r[:n]
	return dA, db, dc

It seems like, when compared to the paper, the code solves M.T @ r = dz for r, whereas the paper solves M.T @ g = -dz for g. So r = -g. But then the equations used in the code to compute (dA, db, dc) seem to match those in the paper, when they should all differ by a negative sign.

Similarly, for the forward-mode derivative, you solve M @ dz = dQ @ pi_z for dz, use the same equations as in the paper despite the sign difference, but you multiply (dx, dy, dz) by -1 before returning, so this is fine.

Is this a sign error in the adjoint derivative, or did I get something wrong?

[sbarratt]

Good question!

The minus sign makes its way to this line:

diffcp/diffcp/cone_program.py

Line 352 in 83080bc

values = pi_z[cols] * r[rows + n] - pi_z[n + rows] * r[cols]

Because otherwise it would be

values = -pi_z[cols] * r[rows + n] + pi_z[n + rows] * r[cols] 

Sorry that it's a bit confusing. We should probably just move it up to the solve line. This minus sign was certainly a pain for us when we were initially debugging.

[spenrich]

I've included some example code (test_diffcp_vjp.txt), where I've just copied in solve_and_derivative_internal and edited the returned DT function to also output r and pi_z. I then run this on the cone program from the readme, explicitly compute -dQ_12.T + dQ_21 (from the paper) with dQ = np.outer(g, pi_z) (where g = -r), and compare this to the value of dA given by solve_and_derivative_internal. For a fixed random seed, I get:

-dQ_12.T + dQ_21 =
[[-2.8563  0.7865 -1.6522  0.0864  1.2953]
 [-0.6479  0.1784 -0.3748  0.0196  0.2938]
 [-1.5848  0.4364 -0.9167  0.0479  0.7187]
 [-0.      0.     -0.      0.      0.    ]
 [ 0.     -0.      0.     -0.     -0.    ]
 [ 1.5824 -0.4357  0.9153 -0.0479 -0.7176]
 [-0.      0.     -0.      0.      0.    ]
 [ 0.     -0.      0.     -0.     -0.    ]
 [ 0.     -0.      0.     -0.     -0.    ]
 [ 0.     -0.      0.     -0.     -0.    ]
 [ 0.     -0.      0.     -0.     -0.    ]]

dA from solve_and_derivative_internal =
[[ 2.8563 -0.7865  1.6522 -0.0864 -1.2953]
 [ 0.6479 -0.1784  0.3748 -0.0196 -0.2938]
 [ 1.5848 -0.4364  0.9167 -0.0479 -0.7187]
 [ 0.     -0.      0.     -0.     -0.    ]
 [-0.      0.     -0.      0.      0.    ]
 [-1.5824  0.4357 -0.9153  0.0479  0.7176]
 [ 0.     -0.      0.     -0.     -0.    ]
 [-0.      0.     -0.      0.      0.    ]
 [-0.      0.     -0.      0.      0.    ]
 [-0.      0.     -0.      0.      0.    ]
 [-0.      0.     -0.      0.      0.    ]]

There seems to be a difference in sign. Am I computing dQ correctly here?

[sbarratt]

It looks like the paper is wrong but the code is right! We're in the process of updating the equations for dA, db, and dc. They should be

Thanks so much for finding this and contacting us about it.